Reconstruction of the solution of inverse Sturm–Liouville problem

In this paper we are concerned with an inverse problem with Robin boundary conditions, which states that, when the potential on [ 0 , 1 / 2 ] and the coefficient at the left end point are known a priori, a full spectrum uniquely determines its potential on the whole interval and the coefficient at the right end point. We shall give a new method for reconstructing the potential for this problem in terms of the Mittag-Leffler decomposition of entire functions associated with this problem. The new reconstructing method also deduces a necessary and sufficient condition for the existence issue.